Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#26080 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×#41×C79 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,22)(12,18)(13,27)(14,31)(15,20)(16,21)(17,24)(19,28)(23,34)(25,33)(26,39)(29,35)(30,40)(32,38)(36,37)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(9,11,14,24,16,25,15,12)(10,17,20,22,21,18,31,33)(13,23,28,37,30,38,29,26)(19,32,27,36,35,34,40,39);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,14)(13,26)(15,25)(16,24)(17,21)(19,34)(20,33)(22,31)(23,28)(27,39)(29,38)(30,37)(32,35)(36,40);(9,13,14,28,16,30,15,29)(10,19,20,27,21,35,31,40)(11,23,24,37,25,38,12,26)(17,32,22,36,18,34,33,39)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 19068);